ECON3350/7350 Cointegration – The Multiple Equation Case
Eric Eisenstat
The University of Queensland
Lecture 11
Eric Eisenstat
(School of Economics) ECON3350/7350 Week 12
1 / 11
VECM Models
Recommended readings
Author
Title
Chapter
Call No
Enders Verbeek
Applied Econometric Time Series, 4e
A Guide to Modern Econometrics
6.3 to end of chapter 9.5
HB139 .E55 2015 HB139 .V465 2012
Eric Eisenstat (School of Economics) ECON3350/7350 Week 12 2 / 11
VAR Representations
Recall the SVAR given by
Γ(L)xt = γ0 + εt, and the VAR given by
A(L)xt = a0 + et, which are related by
A(L) = B−1Γ(L),
E(εtε′t) = Σ, E(ete′t) = Ω,
Ω = B−1Σ B−1′ .
IfA(1)=In−A1−···−Ap isnotinvertible,then
it cannot be the case that all variables in xt are stationary I(0) series; at least one stochastic trend must be present.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 12 3 / 11
Cointegration
Suppose all variables in xt are I(1). Recall that cointegration of rank
r < n means there exist r vectors β1,...,βr such that β′jxt = zt is I(0).
We call the collection of cointegrating vectors β = (β1, . . . , βr), the cointegrating matrix (of dimension n × r).
Recall that cointegrating vectors are not unique: (βκ)′xt = κ′zt is also I(0), so β = βκ for any invertible r × r matrix κ, is also a cointegrating matrix of xt.
Unrestricted β cannot be estimated from the data because only the cointegrating space—spanned by (true) cointegrating vectors
β1, . . . , βr—is identified.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 12 4 / 11
Vector Error Correction Model
The Vector Error Correction Model (VECM) is a generalization of the ECM that we use to analyze cointegrated systems.
Example: n=3,p=2,r=2:
z1,t 1 β21 β31 x1,t x1,t + β21x2,t + β31x3,t
z = β 1 β x2,t= β x +x +β x ∼I(0).
2,t 12 32 x3,t 12 1,t 2,t Then,
xt =a0 +A1xt−1 +A2xt−2 +et, ∆xt =φ0 +Πxt−1 +Ψ1∆xt−1 +et
whereΠ=−A(1)=αβ′ (αis3×2)andΨ1 =−A2.
13 3,t
Eric Eisenstat (School of Economics) ECON3350/7350
Week 12
5 / 11
Relationship Between VAR and VECM
Every VAR can be written as a VECM and vise-versa:
xt =a0 +A1xt−1 +···+Apxt−p +et,
∆xt =a0 +Πxt−1 +Ψ1∆xt−1 +···+Ψp−1∆xt−p+1 +et.
The relationship between VAR coefficients and VECM coefficients is:
Π = −(In − A1 − · · · − Ap), Ψ1 = −(A2 + · · · + Ap),
.
Ψp−1 = −Ap.
In a cointegrated system with r < n the matrix Π is restricted, which
corresponds to non-linear restrictions on A1, . . . , Ap.
To impose explicitly, we need to estimate the VECM.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 12 6 / 11
Granger Representation Theorem
Suppose all elements in xt are either I(1) or I(0) and let r = rank A(1).
1 Ifr=n,thenxt isI(0).
2 If0
————-+————————————————————
_ce1 |
aaugval| 1 . . . . . ajpgval | -1.392016 .32956 -4.22 0.000 -2.0380 -0.7461 abdgval | 1.644154 .37565 4.38 0.000 0.9079 2.3804
aukgval | -0.322184
ausgval | -0.223884
_cons | -78.32383
.26321
.29111
.
-1.22 -0.77 .
0.223 0.441 .
-0.8381
-0.7945
.
0.1937
0.3467
.
————————————————————————–
Eric Eisenstat (School of Economics) ECON3350/7350 Week 12 9 / 11
Estimated Cointegrating Relation Error
Eric Eisenstat (School of Economics) ECON3350/7350 Week 12 10 / 11
Impulse Responses (Cholesky Decomposition)
Eric Eisenstat (School of Economics) ECON3350/7350 Week 12 11 / 11